Two Exact Sequences in Rational Homotopy Theory Relating Cup Products and Commutators

Abstract

Let $X$ be an $(n - 1)$-connected topological space of finite rational type (i.e. ${H_n}(X;Q)$ is finite dimensional over $Q$ for all $n$). Sullivan’s notion of minimal model is used to derive two exact sequences involving the kernel of the cup product operation in dimension $n$ and Whitehead products. The first of these generalizes both a theorem of John C. Wood [JCW] and a theorem of Dennis Sullivan [DS] and states that the kernel of the cup product map ${H^1}(X) \wedge {H^1}(X) \to {H^2}(X)$ is rationally the dual of the second factor of the lower central series of the fundamental group. Other examples are given in the last section.